LIFE  INSURANCE 


LEGAL  NET  VALUES 


A Popular  Treatise 


METHOD  OF  COMPUTING  THE  NET  VALUE  OF 
LIFE  POLICIES  AND  SHOWING  THE 
NATURE  AND  PROPER  USES  OF  THIS  FUND. 


By  GUSTAVUS  W.  SMITH, 

Formerly  Insurance  Commissioner  of  Kentucky. 


NEW  YORK: 

PUBLISHED  BY  THE  SPECTATOR  COMPANY, 


No.  16  Dey  Street. 
1879. 


JK  I ' 


Digitized  by  the  Internet  Archive 
in  2017  with  funding  from 

University  of  Illinois  Urbana-Champaign  Alternates 


https://archive.org/details/lifeinsurancelegOOsmit 


PREFACE. 


The  following  series  of  Articles,  recently  published  in 
‘The  Spectator,”  are  offered  to  the  public  in  pamphlet 
form,  with  the  hope  .that  they  will  be  found  useful  to 
those  who  do  not  already  clearly  understand  the  simple 
elementary  principles  upon  which  life  insurance  calcu- 
lations are  made— after  a table  of  mortality  and  rate  of 
interest  have  been  determined  upon  as  the  bases  for  the 
computations.  These  principles  apply  to  any  table  of 
mortality  and  rate  of  interest.  Without  waiting  the 
completion  of  the  ideal  table,  for  which  the  actuaries 
seem  to  be  still  zealously  searching,  by  means  of  complex 
mathematics  applied  to  statistics  obtained  from  the  ex- 
perience of  companies,  we  have  endeavored  to  explain  the 
method  of  using  the  table  of  mortality  now  designated 
by  law  believing  it  to  be  sufficiently  accurate  for  present 
practical  purposes. 


CONTENTS. 


I.  The  Method  of  Computing  the  Legal  Net  Value  of  a Life  Policy, 
assuming  that  the  Net  Annual  Premiums  and  the  Net  Value  of  a 
life  series  of  Annual  Payments  of  $i  are  known 

II.  The  disposition  that  ought  to  be  made  of  the  Legal  Net  Value  in  case  a 
Renewal  Premium  is  not  paid  when  due.  Interest.  Mortality  Tables. 
The  method  of  calculating  the  net  present  value  of  a life  series  of 
Annual  Payments  of  $i  each 

III.  Net  Cost,  at  any  age,  of  Insurance  for  one  year.  Net  single  premium 

for  whole  life  insurance.  Net  Annnal  Premium  for  whole  Life  In- 
surance. Net  cost  each  year  of  insuring  the  amount  at  risk 

IV.  Tables  I.  to  VI.  inclusive.  The  method  of  determining  that  part  of  the 

net  single  premium  that  will  go  each  year  to  pay  net  cost  of  insuring 

the  amount  at  risk  that  year 

V.  The  method  of  computing  the  Legal  Net  Value  in  case  the  company  has 
contracted  to  furnish  insurance  for  net  premiums,  less  than  those  called 
for  by  the  data  prescribed  by  law.  Comments  upon  net  valuations  . . . 

VI.  The  method  of  determining,  at  any  age,  the  net  present  vaiue  of  $i  to 
be  paid  in  any  designated  number  of  years,  provided  the  person  to 
whom  the  payment  is  to  be  made  is  alive  at  the  end  of  that  time. 

Masseres,  1783. 


LIFE  INSURANCE  LEGAL  NET  VALUES . 


I. 


HE  law  requires  that  a life  insurance  company  shall  hold 


l — invested  in  certain  prescribed  classes  of  securities — 
the  “ net  value”  of  every  policy  it  has  in  force.  A popular  ex- 
planation of  the  simple  principles  upon  which  calculations  of 
these  values  are  based,  will  enable  those  interested  to  form  an 
intelligent  opinion  in  regard  to  the  disposition  that  ought  to  be 
made  of  this  fund  in  case  the  policy  to  which  it  pertains  is  not 
continued  in  force  by  the  payment  of  a premium  when  due.  The 
aggregate  accrued  net  value  of  life  insurance  policies  now  in  force 
in  this  country  is  more  than  three  hundred  million  dollars,  and 
the  amount  is  rapidly  increasing.  The  policyholders  furnish 
the  money  upon  which  this  gigantic  business  is  conducted,  and 
every  policyholder  ought  to  have  some  idea  of  the  practical 
meaning  of  this  law,  which  was  enacted  for  the  government 
of  the  corporation  that  has  the  handling  and  control  of  the 
money  he  pays  to  secure  his  heirs  from  poverty  and  want. 

For  present  purposes  of  illustration  it  is  not  now  necessary 
to  explain  in  detail  the  simple  arithmetical  principles  used  in 
calculating  net  premiums  in  life  insurance.  The  net  annual 
premium  that  will — if  paid  at  the  time  the  policy  issues  and 
payment  is  renewed  at  the  beginning  of  each  succeeding  policy- 
year  that  the  policyholder  is  alive — be  just  sufficient  to  insure 
a given  amount  to  his  heirs  at  the  end  of  any  year  in  which 
he  may  die,  is  readily  susceptible  of  definite  arithmetical 
computation,  based  upon  any  table  of  mortality  and  rate  of  in- 
terest. 

Assuming  that  the  actuaries’  table  of  mortality  and  four  per 
cent  interest  are  designated  by  law  as  the  bases  for  calculating 
the  “ net  value  ” of  a life  insurance  policy,  the  net  annual  pre- 
mium, at  age  20,  to  insure  $1,000  for  life  is  $12.94.  The  net 


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annual  premium  takes  no  account  of  expenses,  profits,  or 
contingencies — these  are  provided  for  by  an  addition  called 
“ loading.” 

The  policyholder,  at  age  20,  pays  the  net  annual  premium 
$12.94,  and  is  insured  for  one  year.  If  he  is  alive  at  age  21 
and  pays  at  that  time  a renewal  net  annual  premium,  $12.94, 
he  is  insured  the  second  year.  But  the  net  annual  premium 
necessary  at  age  21  to  insure  $1,000  for  life  is  $13.27.  The 
insurer  who  accepts  at  age  21  a net  annual  premium  $12.94 
to  insure  $1,000  for  life,  must  have  on  hand,  to  the  credit  of 
this  policy,  an  amount  equivalent  to  the  value  at  this  age  of  a 
life  series  of  annual  payments  each  equal  to  $13.27  less 
$12.94  = $0.33.  The  value  at  age  21  of  a life  series  of  annual 
payments  each  equal  to  $0.33  is  $6.37.  This  $6.37  is  the 
u net  value  ” which  the  law  requires  the  company  to  hold  for 
this  policy  at  age  21.  The  net  annual  premium  which  was  paid 
at  age  20  was  sufficient,  on  the  legal  data,  to  pay  net  cost  of 
insurance  on  this  policy  the  first  year  and  leave  $6.37  in  the 
hands  of  the  company  to  the  credit  of  this  policy  at  the  end  of 
the  year. 

If  this  policyholder  is  alive  at  age  22,  and  pays  at  that  time 
a renewal  net  annual  premium  $12.94,  the  company  can  insure 
him  for  this  net  annual  premium,  notwithstanding  the  fact 
that  the  net  annual  premium  necessary  at  this  age  to  insure 
$1,000  for  life  is  $13.61.  This  can  be  done,  however,  only 
in  case  the  company  holds  at  this  time  to  the  credit  of  this 
policy  an  amount  equal  to  the  value  at  age  22  of  a life  series 
of  annual  payments  each  equal  to  $13.61  less  $12.94  = $0.67. 
The  value  at  age  22  of  a life  series  of  annual  -payments  each 
equal  to  $0.67  is  $12.96.  This  $12.96  is  the  legal  net  value 
of  this  policy  at  the  end  of  the  second  year.  It  was  formed 
partly  by  the  legal  net  value  $6.37  held  by  the  company  to  the 
credit  of  this  policy  at  the  end  of  the  first  year,  and  partly  by  a 
portion  of  the  net  annual  premium  paid  at  the  beginning  of  the 
second  year.  The  other  portion  of  the  second  net  annual 
premium  was  just  sufficient  to  pay  net  cost  of  insuring  this  policy 
during  the  year  between  age  21  and  22. 

In  like  manner  at  the  beginning  of  each  policy-year  that  this 


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policyholder  is  alive  the  net  annual  premium,  $12.94,  will  effect 
his  insurance,  provided  the  company  holds  the  prescribed  legal 
net  value  ; because  this  net  annual  premium  and  this  net  value 
are  together  equivalent  at  each  age  to  the  net  annual  premium 
necessary  at  each  age  to  insure  $1,000  for  life. 

In  case  a policyholder  dies  in  any  year,  the  net  value  of  his 
policy  is  used  by  the  company  in  part  payment  of  his  death  loss  ; 
but  if  the  policyholder  is  alive,  and  continues  his  policy  in  force 
by  the  payment  of  a renewal  premium  at  the  beginning  of  the 
next  policy-year,  the  net  value  is  held  by  the  company,  and  is 
to  be  used  in  part  payment  of  the  policy  to  which  it  pertains 
when  this  policy  matures.  The  net  value  increases  every  year 
that  a policy  continues  in  force  ; at  the  table  limit  of  mortality 
it  is  equal  to  the  amount  of  the  policy.  As  the  net  value  in- 
creases, the  amount  the  company  has  at  risk  on  this  policy 
diminishes  every  year  that  the  policy  continues  in  force,  and  be- 
comes zero  at  the  end  of  the  table  of  mortality.  The  net 
annual  premium  is  composed  of  two  distinct  parts,  one  of  which 
goes  to  pay  net  cost  of  insuring  the  amount  at  risk  each  year, 
the  other  goes  to  form  “ net  value.”  After  a policy  has  been 
in  force  for  a number  of  years  the  net  cost  of  insuring  the 
amount  at  risk  becomes  greater  than  the  net  annual  premium — 
in  this  case  the  balance  of  the  net  cost  is  made  good  from  inter- 
est on  the  net  value  during  the  year — leaving,  however,  enough 
of  this  interest  to  bring  the  net  value  at  the  end  of  the  year  up 
to  the  amount  required  by  law. 

Returning  to  the  policy  for  $1,000  taken  out  at  age  20  ; sup- 
pose this  policyholder  is  alive  at  age  60  and  has  paid  his  annual 
premiums  regularly  every  year.  The  net  annual  premium 
necessary  at  age  60  to  insure  $1,000  for  life  is  $57.55  ; but  the 
company  can,  at  this  age,  insure  this  policyholder  for  the  net 
annual  premium,  $12.94,  because  the  company  is  required  to 
hold  to  the  credit  of  this  policy  an  amount  equal  to  the  value  at 
age  60  of  a life  series  of  annual  payments,  each  equal  to  $57.55 
less  $12.94  = $44-6 1.  The  value  at  age  60  of  a life  series  of 
annual  payments  each  equal  to  $44.61  is  $464.39.  This  is  the 
“ net  value  ” of  this  policy  at  age  60.  This  policyholder  has 
paid  year  by  year,  in  advance,  the  net  cost  of  insuring  the 


8 

amount  at  risk  on  this  policy  each  year,  and  he  has  paid  every 
year — out  of  the  “ loading”  part  of  his  premiums — his  propor- 
tion of  the  expenses,  profits,  and  contingencies.  In  addition,  the 
company  holds  $464.39  to  the  credit  of  this  policy,  which 
amount  enables  the  company  to  continue  the  insurance  upon 
receiving  the  net  annual  premium,  $12.94,  although  the  net 
annual  premium  necessary  at  this  age  to  effect  the  insurance  is 

$57-55- 

Now  suppose  this  policyholder  does  not  pay  his  renewal 
annual  premium  at  age  60,  what  ought  to  be  done  with  the 
$464.39?  Many  companies  settle  this  question  by  so  wording 
the  policy  that  the  insured  agrees  to  forfeit  all  his  interest  in  it 
in  case  he  fails  at  any  time  to  pay  a renewal  premium  when 
due.  This  forfeiture  is  indefensible  in  equity,  and  not  called  for 
by  any  proper  consideration  for  the  rights  of  those  policyholders 
who  continue  to  pay  renewal  premiums. 

The  net  value  of  a life  insurance  policy  at  the  end  of  any 
policy  year  is  as  clearly  a part  of  the  policyholder’s  payments 
for  his  policy  as  the  renewal  premium  then  due  will  be  if  he 
pays  it.  The  net  value  was  formed  out  of  portions  of  the  net 
annual  premiums  he  has  already  paid,  and  for  which  he  has  re- 
ceived no  equivalent.  The  legal  net  value  may,  therefore,  be 
considered  “ unearned  premium.” 

If  there  is  no  legal  contract  to  the  contrary,  this  unearned 
premium — which  the  companies  call  “ reserve,”  but  which  the 
law  calls  “net  value” — should  be  applied  to  the  purchase  of 
paid-up  insurance  for  the  policyholder  who  does  not  continue 
his  original  policy  in  force  by  the  payment  of  a renewal  pre- 
mium when  due. 

Life  insurance  corporations  should  not  be  permitted,  in 
future  contracts,  to  appropriate  to  themselves  the  accrued  legal 
net  value ; but  the  law  should  require  that  paid-up  insurance 
be  issued  for  such  an  amount  as  the  net  value — less  any  in- 
debtedness of  the  policyholder  to  the  company — will  purchase 
at  the  time  a renewal  premium  is  due  and  not  paid.  In  ap- 
plying the  sum  thus  determined  to  the  purchase  of  paid-up  in- 
surance, fair  allowance  should  be  made  for  expenses  and  con- 
tingencies connected  with  the  new  paid-up  policy. 


9 


II. 

THOROUGH  comprehension  of  the  method  of  computing 
these  net  values  requires  a knowledge  of  the  arithmetical 
principles  used  in  calculating  the  present  value,  at  any  age,  of  a 
life’s  series  of  annual  payments  of  $i  each,  and  calculating  the  net 
premium  that  will,  at  any  age,  insure  a given  amount.  Before 
explaining  these  principles  allusion  will  be  made  to  a decision 
of  the  Supreme  Court  of  the  United  States,  which  bears  upon 
the  question  of  the  disposition  that  ought  to  be  made  of  the  ac- 
crued legal  net  value  in  case  a renewal  premium  is  not  paid 
when  due. 

There  were  several  cases  before  the  court  in  which  South- 
ern policyholders  had  during  the  war,  failed  to  pay  renewal 
premiums,  and  had  died  before  the  end  of  the  war.  In  refer- 
ence to  the  legality  of  this  forfeiture  clause  the  court  held  that : 

“ Time  is  material  and  of  the  essence  of  the  contract,  and 
non-payment  at  the  day  involves  absolute  forfeiture,  if  such  be 
the  terms  of  the  contract,  as  is  the  case  here.  Courts  cannot 
with  safety  vary  the  stipulation  of  the  parties  by  introducing 
equities  for  the  relief  of  the  insured  against  their  own  negli- 
gence.” 

This  opinion  settles  the  question  of  the  legality  of  this  for- 
feiture clause  in  life  insurance  policies,  in  cases  where  such  for- 
feiture is  not  prohibited  by  the  charter  of  the  company,  or  by  the 
laws  of  the  State  under  which  the  company  was  incorporated, 
or  of  the  State  in  which  the  insurance  contract  was  made. 
The  court  further  held — 

“ That  such  failure  being  caused  by  a public  war,  without  the 
fault  of  the  assured,  they  are  entitled,  ex  cequo  et  bono , to  re- 
cover the  equitable  value  of  the  policies,  with  interest  from  the 
close  of  the  war.”  * * “In  each  case  the  rates  of  mortality 

and  interest  used  in  the  tables  of  the  company  will  form  the 
basis  of  the  calculation.” 

The  court  illustrated  its  opinion  by  an  example  in  which 
the  policy  is  supposed  to  have  been  issued  at  age  twenty-five, 


10 

for  the  annual  premium  due  at  that  age,  and  the  policyholder 
had  reached  the  age  forty-five  when  he  failed  to  pay  his  renewal 
premium  because  of  a public  war.  On  this  point  the  court  says  : 

“ The  present  value  of  the  amount  assured  is  exactly  repre- 
sented by  the  annuity  which  would  have  to  be  paid  on  a new 
policy,  or  $38  per  annum  in  the  case  supposed,  when  the  party 
is  45  years  old,  while  the  present  value  of  the  premiums  yet  to 
be  paid  on  a policy  taken  by  the  same  person  at  25  is  but 
little  more  than  half  that  amount.  To  forfeit  this  excess,  which 
fairly  belongs  to  the  assured,  and  is  fairly  due  from  the  com- 
pany, and  which  the  latter  actually  has  in  its  coffers,  and  to  do 
this  for  a cause  beyond  individual  control,  would  be  rank  in- 
justice. It  would  be  taking  away  from  the  assured  that  which 
had  already  become  substantially  his  property.  It  would  be 
contrary  to  the  maxim,  that  no  one  should  be  made  rich  by 
making  another  poor.” 

The  court  decided  : — 

First.  u The  policies  in  question  must  be  regarded  as  ex- 
tinguished by  the  non-payment  of  the  premiums,  though  caused 
by  the  existence  of  the  war,  and  that  an  action  will  not  lie  for 
the  amount  insured  thereon.” 

Second.  “ That  the  assured  4 are  entitled  to  recover  the 
equitable  value  of  the  policies  with  interest  from  the  close  of  the 
war.’  ” 

In  other  words,  the  non-payment  of  the  renewal  premium 
having  been  caused  by  war,  this  forfeiture  clause  became  inop- 
erative, and  the  case  was  decided  as  if  this  condition  was  not 
in  the  contract.  The  forfeiture  of  the  accrued  legal  net  value 
has  no  foundation  in  equity,  and  holds  good  against  the  assured 
only  because  the  contract  calls  for  it.  The  original  policy 
ought  not  to  continue  in  force  if  a renewal  premium  is  not  paid 
when  due  ; but  these  corporations,  created  by  State  authority, 
for  the  purpose  of  executing  certain  important  trusts  for  widows 
and  orphans,  should  not  be  permitted,  in  future  contracts,  to 
appropriate  to  themselves  the  accrued  legal  net  value  of  any  of 
their  policies. 

In  calculating  the  net  value  of  a policy  we  need  to  know 
the  net  annual  premium  that  will,  on  the  legal  data,  effect  the 
insurance  at  the  age  of  the  policyholder  at  the  time  for  which 


11 

the  policy  is  being  valued,  and  to  subtract  from  this  the  net 
annual  premium  due  to  the  age  of  the  policyholder  at  the  time 
his  policy  was  issued.  This  gives  the  difference  between  the 
net  annual  premium  required  at  the  time  for  which  the  policy 
is  being  valued  and  the  net  annual  premium  the  insured  will 
pay.  The  question  then  arises  : What  is  the  present  value,  at 
the  time  for  which  this  policy  is  being  valued,  of  a life  series  of 
annual  payments  each  equal  to  this  difference  ? 

Interest . — In  making  these  computations,  the  following 
arithmetical  rule  for  determining  the  present  value  of  $i,  pay- 
able at  the  end  of  any  given  number  of  years,  finds  constant 
application.  “ The  amount  that  will,  at  any  named  rate  of  in- 
terest per  annum,  become  $i  in  one  year  is  obtained  by  dividing 
$i  by  unity  plus  the  rate  of  interest.  The  amount  that  will,  at 
this  rate  of  interest,  compounded  annually,  become  $i  in  any 
designated  number  of  years  is  obtained  by  raising  the  amount 
that  will  become  $i  in  one  year  to  a power  the  exponent  of 
which  is  the  number  of  years.”  In  illustration  of  this  rule, 
suppose  the  rate  of  interest  is  4 per  cent  per  annnm.  The 
amount  that  will,  when  increased  by  interest  at  this  rate  for 
one  year,  become  $ 1 is  equal  to  The  amount  that  will,  at 
the  same  rate  of  interest  per  annum,  compounded  annually, 
become  $1  in  two  years  is  equal  to  il  x ^ And  so  on’ 
multiplying  by  r-h  for  each  additional  year  that  interest  is  to  be 
compounded.  These  calculations  have  been  made,  and  the 
results  are  placed  in  tables  which  show  the  amount  that  will, 
if  invested  at  four  per  cent  per  annum  compounded  annually, 
become  one  dollar  in  any  designated  number  of  years,  from  one 
to  one  hundred. 

Mortality  Tables. — The  amount  that  will,  at  the  named  rate 
of  interest,  become  one  dollar  in  a designated  number  of  years 
being  known,  the  second  step  in  these  calculations  will  have 
been  made  when  we  determine  the  number  of  dollars  that  will 
be  required  at  the  end  of  the  designated  number  of  years.  Mor- 
tality tables  enable  us  to  determine  this  number  of  dollars. 
The  actuaries’  table  of  mortality  was  deduced  from  many 
years’  experience  of  seventeen  principal  life  insurance  com- 
panies of  Great  Britain.  This  table  shows  that  out  of  one  hun- 


12 

dred  thousand  persons  living  at  age  ten,  the  number  of  these 
that  will  die  between  age  ten  and  eleven  is  676.  This  leaves 
99,324  living  at  age  eleven  ; out  of  which  number  674  will  die 
between  age  eleven  and  age  twelve.  And  so  on,  the  table 
shows  the  number  that  will  be  living  at  each  age,  and  the  num- 
ber of  these  that  will  die  before  an  age  one  year  greater. 

A Life  Series  of  Annual  Payments  of  One  Dollar  Each. — 
The  Actuaries’  table  of  mortality  and  four  per  cent  interest  be- 
ing designated  as  the  bases  upon  which  calculations  of  net 
values  in  life  insurance  shall  be  made,  the  net  present  value, 
at  any  age,  of  a life  series  of  annual  payments  of  one  dollar 
each  is  found  as  follows : Suppose  the  age  is  40,  and  that  each 
of  the  78,653  persons  living  at  this  age,  as  shown  by  the  tables 
of  mortality,  are  to  receive  one  dollar  annually  for  life,  the  first 
payment  being  immediate.  The  insurer  will  require  $78,653 
in  hand  in  order  to  enable  him  to  make  the  first  payment.  The 
table  shows  that  of  those  living  at  age  40  there  will  be  77,838 
alive  at  age  41.  Therefore  the  insurer  will  require,  at  the 
beginning  of  the  second  year,  $77,838  in  order  to  pay  at  that 
time  one  dollar  to  each  of  those  that  will  then  be  alive.  The 
amount  that  will,  at  four  per  cent,  become  one  dollar  in 
one  year  is  expressed  by  AL.  Therefore,  77,838  is  the  net 
amount  the  insurer  ought  to  receive  at  the  time  the  contract  is 
entered  into,  to  enable  him  to  pay,  at  the  beginning  of  the  sec- 
ond year,  one  dollar  to  each  of  those  that  will  then  be  alive. 

In  like  manner  we  find  the  net  amount  the  insurer  ought  to 
receive  at  the  time  the  contract  is  entered  into,  in  order  that  he 
may  have  at  the  beginning  of  each  succeeding  year  $1  for 
each  of  those  that  will  then  be  alive.  The  sum  of  these  yearly 
values  to  the  table  limit  of  age  is  the  net  amount  that  will,  if 
paid  in  hand  at  age  40,  enable  the  insurer  to  pay  $1  annually 
for  life  to  each  of  the  78,653  persons  living  at  age  40.  The 
seventy-eight  thousand  six  hundred  and  fifty-third  part  of  this 
amount  is  what  each  one  ought  to  pay.  These  calculations 
have  been  made,  and  they  show  that  the  net  present  value  at 
age  40  of  a life  series  of  annual  payments  of  $1  each  is  $16.09. 
Similar  calculations  have  been  made  at  each  age,  and  the  results 
are  placed  in  tables. 


13 


III. 

NET  Cost  of  Insurance  for  One  Tear . — Suppose  the  age 
of  the  insured  is  40.  The  net  amount  that  will  insure 
$1000  to  be  paid  to  his  heirs  at  the  end  of  the  year,  in  case  he 
dies  between  age  40  and  41,  is  computed  as  follows : The  table 
of  mortality  shows  that  out  of  78,653  persons  living  at  age  40 
the  number  that  will  die  between  age  40  and  41  is  815.  In  case 
the  whole  78,653  living  at  age  40  are  insured  for  $1000  each 
for  one  year,  the  insurer  will  require  $815,000  at  the  end  of  the 
year  to  enable  him  to  pay  $1000  at  that  time  to  the  heirs  of 
each  of  those  that  die  in  that  year.  The  amount  that  will,  at  four 
percent,  become  $1  in  one  year  is  expressed  by  • therefore,  the 
amount  the  insurer  ought  to  receive  at  the  beginning  of  the 
year  is  expressed  by  2L  x 815,000.  Divide  this  by  the  whole 
number  living  at  age  40  and  we  have  the  net  amount  each  ought 
to  pay.  Performing  the  operations  indicated  we  find  this  amount 
is  $9.96. 

In  a similar  manner  the  calculations  have  been  made  at  the 
other  ages,  and  the  results  are  placed  in  tables  which  show  the 
net  amount  that  will  at  each  age  insure  $1000  for  one  year. 

Net  Single  Premium,  that  will  Insure  $1000  for  Life. — 
Again  assume  that  the  age  of  the  insured  at  the  time  the 
policy  is  issued  is  40.  We  have  already  found  the  net  amount 
that  will,  if  paid  at  age  40,  insure  $1000  for  one  year.  To 
find  the  net  amount  that  will  at  age  40  insure  $1000  to  be 
paid  to  the  heirs  of  the  insured  at  the  end  of  the  second  year 
in  case  he  dies  between  age  41  and  age  42  ; obtain  from  the 
table  of  mortality  the  number  of  deaths  between  age  41  and  age 
42 — this  number  is  826.  Therefore,  the  insurer  will  require 
$826,000  at  the  end  of  two  years,  in  order  to  pay  at  that  time 
$1000  to  the  heirs  of  each  of  those  that  will  die  between  age  41 
and  age  42.  The  amount  that  will  at  four  per  cent  per  annum, 
compounded  annually,  become  $1  in  two  years  is  expressed  by 
-I-  Therefore  the  amount  the  insurer  ought  to  receive,  at 

1.04  1.04*  O 7 


14 

the  time  the  contract  is  entered  into,  to  enable  him  to  pay  at  the 
end  of  two  years  $1000  to  the  heirs  of  each  of  those  that  will 
die  between  age  41  and  age  42  is  expressed  by  JL  _L  x 826,- 
000.  Divide  this  by  the  whole  number  living  at  age  40  and  we 
have  the  net  amount  each  ought  to  pay  at  age  40  in  order 
to  insure  $1000  to  his  heirs  in  case  of  his  death  between  age  41 
and  age  42. 

In  a similar  manner  we  compute  the  net  amount  that  will  at 
age  40  insure  $1000  in  case  the  insured  dies  between  age 
42  and  age  43,  and  so  on  for  each  year  to  the  table  limit. 
It  will  be  noticed  that  the  table  of  mortality  is  arranged  by 
years — interest  is  at  a certain  rate  per  cent  per  annum  com- 
pounded annually — the  calculations  are  made  for  insurance  in 
each  separate  year  without  reference  to  insurance  in  any  other 
year,  and  these  separate  yearly  insurances  may  or  may  not  be 
combined,  depending  upon  the  agreement  between  the  insurer 
and  the  insured.  When  these  separate  yearly  insurances  for 
every  year  to  the  table  limit  of  age  are  added  together  their 
sum  is  called  the  net  single  premium  at  age  forty  for  whole  life 
insurance  of  $1000.  These  calculations  have  been  made  and 
the  result  shows  that  at  age  40  the  net  single  premium  that  will 
insure  $1000  for  life  is  $381.04. 

In  a similar  manner  these  calculations  have  been  made  at 
each  age,  and  the  results  are  shown  in  tables. 

The  Net  Annual  Premium. — In  illustration,  again  assume 
that  the  age  is  40.  We  have  just  seen  that  at  this  age  the  net 
single  premium  that  will  insure  $1000  for  life,  is  $381.04  ; and 
we  have  previously  found  that  the  net  value  at  age  40  of  a life 
series  of  annual  payments  of  $1  each  is  $16.09.  These  data 
furnish  the  means  for  forming  a proportion,  the  fourth  term 
of  which,  when  the  proportion  is  solved,  will  show  the  amount 
of  the  net  annual  premium.  The  proportion  is  as  follows  : 

$16.09  to  $3^I-°4  as  $1  is  to  the  fourth  term. 

Solving  this  proportion,  we  find  the  fourth  term  is  $23.67. 
This  is  the  net  annual  premium  that  will,  at  age  40,  insure 
$1000  for  life.  This  is  so,  because  a life  series  of  annual  pay- 
ments of  $1  each,  beginning  at  age  40,  is  equivalent  to  $16.09 
in  hand  at  that  age  ; and  the  proportion  shows  that  this  being 


15 

true,  $23.67  paid  annually  for  life,  beginning  at  age  40,  is  the 
equivalent  of  $381.04  in  hand  at  that  age.  But  $381.04  paid 
at  age  40  will  insure  $1000  for  life;  therefore,  its  equiva- 
lent in  annual  payments  of  $23.67  each  will  insure  $1000  for 
life. 

In  a similar  manner  the  calculations  have  been  made  at  each 
age,  and  the  results  are  shown  in  tables. 

Net  Cost  of  Insuring  the  Amount  at  Risk. — The  legal  net 
value  of  an  ordinary  life  policy,  at  the  end  of  any  policy  year, 
is  the  value  at  that  time,  of  a life  series  of  annual  payments 
each  equal  to  the  difference  between  the  net  annual  premium 
due  to  the  age  at  which  the  policy  is  being  valued,  and  the 
net  annual  premium  the  insured  pays.  The  amount  the  com- 
pany has  at  risk  any  year  is  the  amount  of  the  policy  less  the 
legal  net  value  at  the  end  of  that  year.  Having  calculated  the 
legal  net  value  at  the  end  of  any  policy  year  the  amount  the 
company  has  at  risk  during  that  year  becomes  known.  It  is 
now  proposed  to  illustrate  the  method  by  which  we  determine 
what  part  of  each  net  annual  premium  goes  to  pay  net  cost  of 
insurance,  and  what  part  goes  to  form  legal  net  value. 

In  illustration : assume  that  an  ordinary  whole  life  policy  for 
$1000  is  taken  out  at  age  42,  the  net  annual  premium  at  this 
age  (actuaries’  table  of  mortality  and  4 per  cent  interest)  is 
$25,554.  (See  table.)  The  legal  net  value  of  this  policy  at 
the  end  of  the  first  year,  computed  by  the  method  just  referred 
to,  is  $158.51.  The  amount  at  risk  on  this  policy  during  the 
year  between  age  42  and  43  is  $1000  less  $158.51  = $984,149. 
The  table  shows  that  at  age  42,  the  net  amount  that  will  insure 
$1000  for  one  year  is  $10,476. 

From  this  we  find  that  at  age  42,  the  net  amount  that  will  in- 
sure $984,149  for  one  year  is  $10.31.  Subtract  this  from  the 
net  annual  premium  $25,554,  Pa*d  at  age  42,  and  we  have 
$15,244  of  this  premium  left  after  providing  net  cost  of  insuring 
the  amount  at  risk  during  the  year.  This  $15,244,  increased  at 
4 per  cent  will  become  $15.85  at  the  end  of  the  year.  This 
$15.85  is  the  legal  net  value  at  the  end  of  the  year,  as  calcu- 
lated by  the  method  above  referred  to. 

In  like  manner  that  part  of  the  net  annual  premium  at  age  43 


16 

which  pays  net  cost  of  insurance  on  the  amount  at  risk  during 
the  year  between  age  43  and  age  44  may  be  determined. 
When  we  have  found  this  amount  its  value  at  age  42  is  obtained 
by  multiplying  it  by  Bear  in  mind  that  the  premium  at 
age  43  is  to  be  paid  only  in  case  the  insured  is  then  alive. 
Therefore,  the  above  amount  must  be  multiplied  by  the  num- 
ber living,  as  shown  by  the  table  at  age  43,  and  this  product 
divided  by  the  number  living  at  age  42,  in  order  to  determine 
the  net  present  value* — at  the  time  this  policy  is  issued — of  that 
part  of  the  net  annual  premium  at  age  43,  which  will  pay  net 
cost  of  insuring  the  amount  at  risk  on  this  policy  during  the 
year  between  age  43  and  age  44.  These  calculations  are  made 
in  a similar  manner  for  each  year  to  the  table  limit  of  age. 

The  net  present  value,  at  age  42,  of  u the  normal  contribu- 
tions this  policyholder  is  liable  to  have  to  make  in  payment 
of  death  claims  other  than  his  own”  is  obtained  by  find- 
ing the  sum  of  the  foregoing  amounts  for  each  year  to  the 
table  limit.  This  sum  will,  if  paid  at  age  42,  insure  the 
amount  at  risk  on  this  policy  every  year  to  the  table  limit.  That 
part  of  the  $1000  insurance  not  included  each  year  in  the 
amount  at  risk  is  provided  for  by  the  legal  net  value  at  the  end 
of  that  year. 

In  a manner  similar  to  that  indicated  above,  for  age  42,  the 
net  value  of  these  “ contributions  ” may  be  calculated  for  any 

age- 

It  is  important  that  every  policyholder,  in  a life  insurance 
company,  should  know  that  the  contribution  he  is  liable  to 
have  to  make  in  payment  of  death  claims,  other  than  his  own, 
is  balanced  by  the  liability  of  the  other  policyholders  to  con- 
tribute to  his  death  claim.  These  other  policyholders  are  not 
entitled  to  the  contributions  he  is  liable  to  make  in  payment 
of  their  death  claims  unless  their  liability  to  have  to  contribute 
to  his  death  claim  is  clearly  acknowledged.  This  principle 
will  be  again  referred  to  in  another  connection. 


* See  Article  VI. 


17 


IV. 

THE  method  by  which  we  compute  the  amount  that  will, 
when  increased  by  interest  at  any  designated  rate  per 
annum,  compounded  annually,  become  one  dollar  in  a given 
number  of  years  has  already  been  stated. 


INTEREST. 

Table  I — Shows  the  amount  that  will,  when  increased  at  4 
per  cent  per  annum,  compounded  annually,  become  $1  in  any 
designated  number  of  years,  from  one  to  one  hundred,  in- 
clusive : 


J 

Years. 

Amount  at 
\per  cent. 

Years. 

Amount  at 

4 per  cent. 

Years. 

Amount  at 

4 per  cent. 

Years. 

Amount  at 

4 per  cent. 

1 

$0.9615385  1 

26 

$0.3606892 

5i 

$0  1353006 

76 

$0.0507535 

2 

0.9245562 

27 

0.3468166 

52 

0.1300967 

77 

0.0488015 

3 

0.8889964 

28 

0.3334775 

i 53 

0.1250930 

78 

0.0469245 

4 

0 8548042 

j 29 

0.3206514 

i 54 

0.1202817 

79 

0.0451197 

5 

0.8219271 

30 

0.3083187 

55 

0.1156555 

80  " 

0.0438843 

6 

0.7903145 

3i 

0 2964603 

56 

0. 1 1 12072 

81 

0.0417157 

7 

0.7599178 

32 

0.2850579 

. 57 

0. 1069300 

82 

0.0401 1 12 

8 

0.7306902 

33 

0.2749242 

58 

0.1028173 

83 

0.0385685 

9 

0.7025867 

34 

0 2635521 

59 

0.0988628 

84 

0 0370851 

10 

0.6755642 

35 

0.2534155 

! 60 

0.0950604 

85 

0.0356587 

11 

0.6495809 

36 

0.2436687 

61 

0 0914042 

86 

0.0342873 

12 

0.6245970 

37 

0.2342968 

62 

0 0878887 

87 

0.0329685 

13 

0 6005741 

38 

0.2252854 

63 

0.0845083 

88 

0.0317005 

14 

0.5774751 

39 

0.2166206 

64 

0.0812580 

89 

0.0304812 

15 

0.5552645 

40 

0.2082890 

65 

0.0781327 

90 

0.0293080 

16 

0.5339082 

4i 

0.2002779 

66 

0.0751276 

9i 

0.0281816 

1 7 

0.5133732 

42 

0.1925749 

67 

0.0722381 

92 

0 0270977 

18 

0.4936281 

43 

0.1851682 

68 

0.0694597 

93 

0.0260555 

19 

0.4746424 

44 

0 1780463 

69 

0.0667882 

94 

0.0250534 

20 

0.4563869 

45 

0.1711984 

70 

0.0642194 

95 

0.0240898 

21 

0.4388336 

46 

0 1646130 

7i 

0.0617494 

96 

0.0231632 

22 

0.4219554 

47 

0.1582826 

72 

0.0593744 

97 

0.0222723 

23 

0.4057263 

48 

0.1521948 

73 

0.0570908 

98 

0.0214357 

24 

0 3901215 

49 

0.1463411 

74 

0.0548950 

99 

0.0205920 

25 

0 3751168 

50 

0.1407126 

75 

0.0527837 

100 

0.0198000 

18 

ACTUARIES*  TABLE  OF  MORTALITY. 

Table  II — This  table  was  compiled  from  observation  and 
experience,  as  previously  explained.  It  shows  that  out  of 
ioo.ooo  insured  persons  living  at  age  io,  the  number  of  these 
that  will  die  between  age  io  and  age  n is  676 ; the  number 
that  will  die  between  age  11  and  age  12  is  674 ; and  so  on  giv- 
ing the  number  that  will  die  each  year  to  the  table  limit  of 
age.  The  table  also  shows  the  number  that  will  be  living  at 
each  age. 


Age. 

Living. 

Deaths. 

Age. 

Living. 

Deaths. 

Age. 

Living. 

Deaths. 

10 

100,000 

676 

41 

78,653 

815 

70 

35,837 

2.327 

11 

99.324 

674 

4i 

77.838 

826 

7i 

33,5io 

2,351 

12 

98.65  - 

672 

42 

77  012 

839 

72 

3i  159 

2 362 

13 

97978 

671 

43 

76,173 

857 

73 

28,797 

2,358 

14 

97.307 

671 

44 

75.316 

881 

74 

26,439 

2,339 

IS 

96.636 

671 

45 

74.435 

909 

75 

24,100 

2,303 

16 

95.965 

672 

46 

73,526 

944 

76 

21,797 

2,249 

1 7 

95.293 

673 

47 

72.582 

981 

77 

19,548 

2,179 

18 

94,620 

675 

48 

71,601 

1,021 

78 

17.369 

2,092 

19 

93  94- 

677 

49 

70  580 

1,063 

79 

15.277 

1,987 

20 

93,268 

680 

50 

69.517 

1,108 

80 

13,290 

1,866 

21 

92,588 

683 

5i 

68,409 

1,156 

81 

11,424 

1,730 

22 

91.905 

686 

52 

67,253 

1,207 

82 

9.694 

1,582 

23 

91,219 

690 

53 

66,046 

1,261 

83 

8,112 

1,427 

24 

90.529 

694 

54 

64,785 

1,316 

84 

6,685 

1,268 

25 

89,835 

698 

55 

63,469 

1.375 

85 

5,4i7 

1, hi 

26 

89,137 

703 

56 

62,094 

1,436 

86 

4-306 

958 

27 

88,434 

708 

57 

60,658 

1,497 

87 

3,348 

811 

28 

87,726 

714 

58 

59.i6i 

1,561 

88 

2 537 

673 

29 

87,012 

720 

59 

57,6-0 

1,627 

89 

1 864 

545 

30 

86,292 

727 

60 

55  973 

1,698 

90 

1.319 

427 

3i 

85.565 

734 

61 

54  275 

1,770 

91 

892 

322 

32 

84.831 

742 

62 

52  505 

1,844 

92  ! 

. 570 

231 

33 

84,089 

750 

63 

50,661 

i,9I7 

93 

339 

155 

34 

83.339 

758 

64 

48,744 

1,990 

94  1 

184 

95 

35 

82,581 

767 

65 

46,754 

2 061 

95 

89 

52 

36 

81,814 

776 

66 

44.693 

2,128 

96 

37 

24 

37 

81,038 

785 

67 

42,565 

2,191 

97 

13 

9 

38 

80,253 

795 

68 

40.374 

2,246 

98 

4 

3 

39 

79,458 

805 

69 

38,128 

2,291 

99 

1 

1 

The  methods  used  in  computing  the  values  shown  in  the 
following  tables  have  already  been  explained  and  illus- 
trated : 


19 


Table  III — Shows  the  net  value  at  each  age  of  a life  series 
of  annual  payments  of  $i  each,  the  first  immediate — Actuaries* 
Table  of  Mortality — interest  at  four  per  cent. 


Years . 

Amount  at 
\per  cent. 

Years . 

Amount  at 

4 per  cent. 

Years. 

II 

Amount  at 

4 per  cent. 

Years. 

Amount  at 
\per  cent. 

10 

$20.4536 

33 

$17.5196 

56 

$11.6698 

79 

$4 . 8986 

11 

20.3694 

34 

17.335° 

57 

H-3593 

80 

4.6607 

12 

20.2818 

35 

17.1443 

58 

11.0463 

81 

4.4290 

13 

20 . 1907 

36 

16.9476 

59 

10. 7311 

82 

4.2026 

14 

20.0959 

37 

16 . 7443 

60 

10.4147 

8J 

3.9802 

15 

19.9976 

38 

16 . 5342 

61 

10.0977 

84 

3.7611 

16 

I9-89S7 

39 

16.3172 

62 

9-78o5 

85 

3-5436 

17 

19.7901 

40 

16 . 0929 

63 

9.4641 

86 

3-3279 

18 

19 . 6807 

4i 

15 . 8610 

64 

9.1489 

87 

3-1138 

19 

I9-5675 

42 

15.6212 

65 

8 8356 

88 

2 . 9012 

20 

19.4504 

43 

15 . 3736 

66 

8.5248 

89 

2.6911 

21 

I9-3293 

44 

15.1186 

67 

8 . 2170 

90 

24854 

22 

19 . 2042 

45 

14.8571 

68 

7.9130 

9i 

2.284? 

23 

19.0747 

46 

14.5896 

69 

7.6130 

92 

2.0902 

24 

18.9410 

47 

14..  3170 

70 

7.3172 

93 

1 . 9065 

25 

18.8027 

48 

14..  0304 

7i 

7 . 0261 

94 

1.7369 

26 

18.6598 

49 

13 . 7572 

72 

6.7400 

95 

1 5843 

27 

18.5122 

50 

13  4/703 

73 

6-4593 

96 

1.4618 

28 

18.3597 

5i 

* TV 

13 . 1792 

74 

6. 1840 

97 

1.3670 

29 

18  2022 

52 

12.8841 

75 

5-9I46 

98 

1 . 2404 

30 

18.0397 

53 

12. ^8^3 

76 

5-6512 

99 

1. 0000 

3i 

17.8718 

54 

• JWJJ 

12 . 2832 

77 

5-3938 

32 

17-6985 

55 

11.9779 

78 

5.1428 

Table  IV — Shows  the  net  cost  of  insuring  $1000  for  one 
year,  at  different  ages,  from  20  to  70,  inclusive.  (Actuaries* 
Table  of  Mortality — interest  at  four  per  cent.) 


Age. 

Age. 

J 

Age. 

20 

$7,010 

37 

$9,314 

54 

$i9-532 

21 

7093 

38 

9 525 

55 

20.831 

22 

7.177 

39 

9.741 

56 

22.237 

23 

7-273 

40 

9963 

57 

23.730 

24 

7-371 

4i 

10.204 

58 

25  371 

25 

7-471 

42 

10.476 

59 

27.160 

26 

7-583 

43 

10.818 

60 

29.169 

27 

7.698 

44 

11.247 

61 

31-357 

28 

7.826 

45 

11.742 

62 

33-770 

29 

7-956 

46 

12.345 

63 

36.384 

30 

8.101 

47 

12.996 

64 

39-255 

3i 

8.248 

48 

13.711 

65 

42.386 

32- 

8.410 

49 

14.482 

66 

45.782 

33 

8.576 

5o 

15-326 

67 

49.494 

34 

8.746 

5i 

16.248 

68 

53-490 

35 

8931 

52 

17257 

69 

57776 

36 

9.122 

53 

18,359 

70 

62.436 

20 


Table  V— Shows  the  net  single  premium  that  will,  at  differ- 
ent ages,  from  20  to  70,  inclusive,  insure  $1000  for  life.  (Actu- 
aries’ Table  of  Mortality — interest  at  four  per  cent.) 


Age. 

Age. 

Age. 

20 

$251,907 

37 

$355-989 

54 

$527,567 

. 21 

256.564 

38 

364.065 

55 

539-312 

22 

261.377 

39 

372.414 

56 

55I-I57 

23 

266  357 

40 

381.040 

57 

563-103 

24 

271  500 

4i 

389.960 

58 

575-142 

25 

276.8x6 

42 

399-I83 

59 

587257 

26 

282.312 

43 

408.709 

60 

599-433 

2 7 

287,990 

44 

418.515 

61 

611.628 

28 

293.856 

45 

428.571 

62 

623  826 

29 

299.913 

46 

438.862 

63 

635-995 

3° 

306.168 

47 

449  346 

64 

648  120 

31 

312  624 

48 

460.022 

65 

660.171 

32 

319.289 

59 

470.878 

66 

672.124 

33 

326.167 

50 

481.906 

67 

683.968 

34 

333 -267 

5i 

493.107 

68 

695-654 

35 

340.600 

52 

504.460  | 

69 

707  192 

36 

348  170 

53 

5I5-949 

70 

718.569 

Table  VI — Shows  the  net  annual  premium  that  will,  at 
different  ages,  from  20  to  70,  inclusive,  insure  $1000  for  life. 
(Actuaries’  Table  of  Mortality— interest  at  four  per  cent.) 


Age. 

Age. 

Age. 

20 

$12,948 

37 

$21  260 

54 

$42,950 

21 

I3-273 

38 

22  018 

55 

45-025 

22 

13  610 

39 

22  823 

56 

47  230 

23 

24 

13- 963 

14- 334 

40 

41 

23.677 

24.586 

57 

58 

49-571 

1 52  067 

25 

14.722 

42 

25  554 

59 

54  724 

26 

15.129 

43 

26.585 

60 

57-556 

27 

28 

15-557 

16  005 

44 

45 

27.682 

28  845 

61 

62 

60.572 

63.782 

29 

16  477 

46 

30.080 

63 

67.199 

3° 

16.972 

47 

3I-385 

64 

70.841 

31 

17.492 

48 

32  767 

65 

74.718 

32 

18.040 

49 

34  227 

66 

78.846 

33 

18.616 

50 

35  775 

67 

83-237 

34 

19  225 

5i 

37-415 

68 

87-9t3 

35 

19  866 

52 

39.I5I 

69 

92  892 

36 

20544 

53 

40.996 

70 

98  202 

21 

The  tables  given  above  present  an  apparently  formidable 
array  of  figures,  but  on  examination  it  will  be  found  that  all 
the  tables  are  arranged  by  successive  whole  years,  and  that 
opposite  each  age  or  number  of  years  will  be  found  in  its  pro- 
per column  certain  specific  information. 

In  illustration  take  age  30.  Table  I shows  that  $0.3083187 
will,  at  4 per  cent,  compounded  annually,  become  $1  in  30 
years.  Table  II  shows  that  at  age  30  there  are  86,292  per- 
sons living  out  of  100,000  that  were  living  at  age  10 ; and  that 
out  of  this  number  living  at  age  30  the  number  that  will  die 
before  age  31  is  727.  It  also  shows  that  out  of  the  100,000 
living  at  age  10  the  number  of  these  that  will  die  between  age 
30  and  age  31  is  727.  Table  III  shows  that  at  age  30  the  net 
present  value  of  a life  series  of  annual  payments  of  $1  each  is 
$18.0397.  Table  IV  shows  that  at  age  30  the  net  cost  of 
insuring  $1000  for  one  year  is  $8,101.  Table  V shows  that 
at  age  30  the  net  single  premium  that  will  insure  $1000  for  life 
is  $306,168.  This  is  for  paid  up  insurance.  Table  VI  shows 
that  at  age  30  the  net  annual  premium  that  will  insure  $1000 
for  life  is  $16,972. 

In  like  manner,  we  can  readily  find  in  the  tables  information 
similar  to  the  above,  by  looking  in  the  proper  column,  opposite 
to  the  designated  number  of  years  or  age  of  the  insured.  * 

In  illustration : Suppose  that  it  is  required  to  find  the  legal 
net  value  at  age  60  of  a whole  life  policy  for  $1000  issued  at 
age  20. 

The  net  annual  premium  necessary  at  age  60  to  insure  $1000 
for  life  is  $57*556  (see  table  VI).  But  the  policyholder  who 
took  out  his  policy  at  age  20  pays  a net  annual  premium  of 
$12,948  only  (see  same  table).  The  difference  between  the 
net  annual  premium  necessary  at  age  60  to  effect  this  insurance, 
and  the  net  annual  premium  the  insured  pays  at  this  age,  is 
$57,556  less  $12,948  = $44,608.  The  value  at  age  60  of  a 
life  series  of  net  annual  premiums  each  equal  to  $44,608  is 
obtained  by  using  table  III,  from  which  we  find  the  value  at 
age  60  of  a life  series  of  annual  payments  of  $1  each  is 
$10.4147.  Multiply  this  by  $44,608  and  we  have  $464,578, 
which  is  the  net  value  at  age  60  of  a life  series  of  annual  pay 


22 

ments,  each  equal  to  the  difference  between  the  net  annual 
premium  necessary  at  age  60  to  insure  $1000  for  life,  and  the 
net  annual  premium  the  insured  will  pay.  This  $464,578  is 
the  legal  net  value  of  this  policy  at  age  60.  And  this  is  the 
amount  the  law  requires  the  company  to  hold  to  the  credit  of 
this  policy  at  this  age. 

If  by  the  terms  of  the  contract  the  insured  was  not  required 
to  pay  any  more  net  annual  premiums — in  other  words,  if  this 
policy  was  full  paid  at  age  60 — the  legal  net  value  would  then 
be  the  net  present  value  at  age  60  of  a life  series  of  annual  pay- 
ments, each  equal  to  $57,556.  The  net  present  value  at  age  60 
of  a life  series  of  annual  payments  of  $1  each  is  $10.4147 
(see  table  III).  Multiply  this  by  $57,556,  and  we  have 
$599.43,  which  is  the  net  single  premium  at  age  60. 

Net  cost  each  year  of  insuring  the  amount  at  risk . — Sup- 
pose a paid-up  whole  life  policy  for  $1000  is  issued  at  age  20. 
The  net  single  premium  in  this  case  is  $25 1 .907  (see  table  V),  and 
this  is  the  net  amount  that  will,  if  paid  at  age  20,  insure  $1000  to 
the  heirs  of  the  insured  at  the  end  of  any  year  in  which  he  may 
die.  Table  V shows  that  the  net  single  premium  at  age  21 
necessary  to  insure  $1000  for  life  is  $256,564.  The  net  single 
premium  ($251,907)  paid  at  age  20,  will  pay  net  cost  of  insuring 
the  amount  at  risk  on  this  policy  during  the  year  between  age  20 
and  age  21,  and  leave  in  the  hands  of  the  company  at  the  end  of 
the  year  the  net  single  premium  $256,564,  which  is  the  net 
amount  requisite  at  this  age  to  insure  $1000  for  life. 

Since  this  is  a paid-up  policy  its  legal  net  value  af^age  21  is 
the  net  single  premium  that  will  at  that  age  insure  $1000  for 
life.  If  the  policy  continues  in  force  the  company  must  hold 
at  the  end  of  the  year  the  net  single  premium,  $256,564,  after 
net  cost  of  insuring  the  amount  at  risk  during  the  year  has  been 
provided  for  out  of  the  net  single  premium  paid  at  the  begin- 
ing  of  the  year.  The  amount  at  risk  on  this  policy  during  the 
year  between  age  20  and  age  21,  is  equal  to  the  amount  of  the 
policy  less  the  legal  net  value  at  the  end  of  the  year.  It  is, 
therefore,  $1000  less  $256,564  = $743,346.  The  net  amount 
that  will  at  age  20  insure  $1000  for  one  year  is  $7  01  (see  table 
IV).  From  this  we  find  that  the  net  amount  that  will,  at  age 


23 

20,  insure  $743,346  for  one  year  is  $5,211.  This  is  the  amount 
necessary  at  age  20  to  insure  the  amount  at  risk  on  this  policy 
during  the  year  between  age  20  and  age  21.  Subtract  this 
from  the  net  single  premium  paid  at  age  20,  and  we  have 
$251,907  less  $5,211  = $246,696,  which  is  that  part  of  the  net 
single  premium  paid  at  age  20  which  goes  to  form  legal  net 
value  for  this  policy  at  the  end  of  the  year.  Increase  $246,696 
by  4 per  cent,  and  we  have  $256,564  at  the  end  of  the  year. 
This  is  the  net  single  premium  at  age  21  (see  table  V).  Each 
policyholder  living  at  age  20  contributes  $5,211  out  of  the  net 
single  premium,  $251,907,  to  pay  death  claims  during  the  year. 
Each  of  those  who  die  between  age  20  and  age  21  contributes 
to  his  own  death  claim  in  addition  to  this  $5,211,  the  legal  net 
value  of  his  policy  at  the  end  of  the  year.  The  contribution, 
$5,211,  made  at  the  beginningof  the  year  by  each  policyholder 
living  at  that  time,  insures  the  amount  at  risk  on  his  policy 
during  the  year ; in  other  words,  pays  net  cost  of  insuring 
$743-346  during  the  year  between  age  20  and  age  21.  The 
legal  net  value  at  the  end  of  the  year  makes  up  the  $1000  death 
claim. 

If  the  policyholder  is  alive  at  age  21  the  company  holds 
$256,564  to  the  credit  of  his  policy,  after  deducting  net  cost  of 
insuring  the  policy  for  the  year  between  age  20  and  age  21.  This 
$256,564  must  provide  net  cost  of  insuring  this  policy  during 
the  year  between  age  21  and  age  22,  and  leave  in  the  hands  of 
the  company  the  legal  net  value  at  the  end  of  the  year.  This 
legal  net  value  is  the  net  single  premium  that  will  at  age  22 
insure  $1000  at  the  end  of  any  year  in  which  the  insured  may 
die.  Table  V shows  that  this  amount  is  $261,377.  The 
amount  at  risk  during  the  year  between  age  21  and  age  22  is 
$1000,  less  $261,377  = $738,623.  Table  IV  shows  that  at 
age  21  the  net  cost  of  insuring  $1000  for  one  year  is  $7,093. 
From  this  it  follows  that  at  age  21  the  net  cost  of  insuring 
$738,623  for  one  year  is  $5,239.  Deduct  this  from  the  net 
single  premium  at  age  21,  and  we  have  $256,564,  less  $5,239  = 
$25I-325’  which  is  that  part  of  the  net  single  premium  at  age 
21  that  goes  to  form  the  legal  net  value  at  the  end  of  the  year. 
Increase  this  $251,325  by  4 percent  and  we  have  $261,378, 


24: 

which  is  the  net  single  premium  necessary  at  age  22  to  insure 
$1000  at  the  end  of  any  year  in  which  the  policyholder  may 
die.  The  part  of  the  net  single  premium  at  age  22  that  will 
pay  net  cost  of  insuring  the  amount  at  risk  on  this  policy,  during 
the  year  between  age  22  and  age  23,  may  be  found  in  a similar 
manner,  and  so  on,  for  each  and  every  year  to  the  table  limit  of 
age,  at  which  time  the  legal  net  value  will,  when  increased  by 
4 per  cent,  amount  to  $1000.  This,  too,  after  this  policyholder 
has  paid  in  advance  each  year  the  net  cost  of  insuring  the 
amount  at  risk  on  his  policy  during  the  year,  and  has  also  paid, 
out  of  the  “ loading”  added  to  the  net  premium,  his  proportion 
of  the  expenses,  profits  and  contingencies  on  his  policy  every 
year  from  the  time  it  was  issued  at  age  20  until  he  has  reached 
age  100. 

Having  found  that  part  of  the  legal  net  value  that  will,  at 
the  beginning  of  each  year,  pay  net  cost  of  insuring  the  amount 
at  risk  on  this  policy  during  that  year,  we  can,  by  using  the 
mortality  table  and  rate  of  interest  designated  by  law,  find  its 
value  at  age  20.  For  instance,  we  found  above  that  $5,239  is 
that  part  of  the  legal  net  value  of  this  policy  at  the  end  of  the 
first  year,  that  will  pay  net  cost  of  insuring  the  amount  at  risk 
on  this  policy  during  the  year  between  age  21  and  age  22.  To 
determine  the  net  value  at  age  20  of  $5,239,  at  age  21,  in  case 
the  insured  is  then  alive,  we  first  find  the  present  value  of  $1 
in  one  year ; this  at  4 per  cent  is  Multiply  yih  by  the 
number  of  persons  shown  by  the  table  of  mortality  to  be  living 
at  age  21,  and  divide  the  product  by  the  number  shown  to  be 
living  at  age  20,  the  result  gives  the  net  value  at  age  20  of  $1 
to  be  paid  at  age  21,  in  case  the  insured  is  then  alive.*  Mul- 
tiply this  result  by  5.239  and  we  have  the  net  value  at  age  20  of 
the  net  cost  of  insuring  the  amount  at  risk  on  this  policy  during 
the  second  year. 

In  a similar  manner  the  net  value  at  age  20  of  the  net  cost 
of  insuring  the  amount  at  risk  on  this  policy  each  year  to  the 
table  limit  of  age  may  be  determined.  The  sum  of  all  these 
yearly  values  gives  the  net  present  value  at  age  20  that  will 
pay  net  cost  of  insuring  the  amount  at  risk  on  this  policy  every 
year  to  the  table  limit. 

* See  Article  VI. 


25 


V. 

WE  have  previously  seen  that,  for  an  ordinary  life  policy 
of  $1000,  issued  at  age  twenty,  the  legal  net  value  at  age 
sixty  is  $464,578.  In  this  case  the  net  annual  premiums, 
paid  and  payable,  are  those  called  for  by  the  Actuaries’  table 
of  mortality  and  4 per  cent  interest,  viz.,  $12,948.  Upon  the 
same  data  the  net  annual  premium  necessary  at  age  60  to  insure 
$1000  for  life  is  $57,556.  To  obtain  the  legal  net  value  of  this 
policy,  at  age  60,  we  first  found  the  difference  between  the  net 
annual  premium  due  to  age  60*  and  that  due  to  age  20  and  then 
multiplied  this  difference  by  the  value  at  age  60  of  a life  series 
of  annual  payments  of  $1  each. 

The  same  result  would  have  been  obtained  by  first  finding 
the  net  single  premium  that  will,  at  age  60,  insure  $1000  for 
life — (which  is  $599.433.,  see  Table  V) — and  then  subtracting 
from  this  net  single  premium,  the  value  at  age  60  of  the  net 
annual  premiums  yet  to  be  paid.  In  case  the  company  has 
contracted  to  furnish  insurance  for  net  premiums  less  than 
those  called  for  by  the  table  of  mortality  and  rate  of  interest 
prescribed  by  the  law,  the  legal  net  value  required  to  be  held 
for  this  policy  by  the  company  will  be  greater  than  that  ob- 
tained above.  For  instance,  suppose  the  net  annual  premiums 
the  company  has  contracted  to  receive  for  this  policy  at  and 
after  age  60,  are  but  $10  each — the  value  at  age  60  of  a life 
series  of  annual  payments  of  $1  each  is  $10.4147  ; therefore, 
the  value  at  the  same  age  of  a like  series  of  annual  payments 
of  $10  each  is  $104,147.  The  net  single  premium — on  the  legal 
data  above  named,  that  will,  at  age  60,  insure  $1000  for  life — 
being,  as  before  stated,  $599,433,  we  have  $599,433  less 
$io4.i47=$495.286=the  legal  net  value  of  this  policy  at 
age  60,  in  case  the  net  annual  premiums  yet  to  be  paid  are  $10 
each.  When  these  premiums  were  $12,948  each,  the  legal  net 
value  at  age  60  was,  as  we  have  seen,  $464,578. 

Voluminous  tables,  showing  the  net  value  of  various  kinds  of 


20 

life  policies  at  the  end  of  each  year  of  the  policy,  have  been 
prepared  and  published  at  great  expense.  “ Valuation  Tables” 
are  very  useful  in  finding  the  liability  of  a company  on  account  of 
the  accrued  legal  net  value  of  the  policies  it  has  in  force.  These 
tables  are  constructed  on  the  assumption  that  the  net  premiums 
receivable  by  the  company  are  those  called  for  by  the  table  of 
mortality  and  rate  of  interest  which  were  the  bases  upon  which 
the  calculations  for  the  tables  were  made.  When  the  net  pre- 
miums the  company  has  contracted  to  receive  are  less  than 
those  called  for,  as  just  stated,  the  tables  show  too  small  a net 
value.  This  must  be  corrected,  in  each  such  case,  by  adding 
to  the  amount  given  in  the  valuation  tables,  a sum  equal  to  the 
value  at  the  age  for  which  the  valuation  is  being  made,  of  a 
series  of  payments  each  equal  to  the  difference  between  the 
net  premium  called  for  by  the  data  upon  which  the  tables  are 
computed,  and  the  net  premium  the  company  has  contracted  to 
receive.  If  the  company  is  to  receive  no  further  net  premiums 
the  legal  net  value  is  the  net  single  premium  necessary  at  that 
time  to  effect  the  insurance. 

If  the  future  net  premiums  payable  are  greater  than  those 
called  for  by  the  State  standard — only  the  value  of  that  portion 
which  is  called  for  by  that  standard  should  be  deducted  from 
the  net  single  premium  in  order  to  determine  the  legal  net 
value.  In  this  case  that  portion  of  the  future  net  premiums 
which  is  in  excess  of  that  called  for  by  the  State  standard  must 
be  treated  as  “loading” — it  has  no  proper  place  in  the  princi- 
ples of  net  valuation,  under  the  law. 

If  the  liability  of  a company,  at  any  time,  on  account  of  the 
net  value  of  the  policies  it  has  in  force,  is  not  accurately  com- 
puted on  the  data  prescribed  by  law,  no  proper  conception  can 
be  formed  of  the  legal  condition  of  the  company  at  that  time. 
What  the  condition  of  the  company  may  probably  be  at  some 
future  time  is  another  question. 

The  law  requires  of  life  insurance  corporations  something 
more  than  bare  commercial  solvency.  It  is  not  sufficient  that 
probable  future  profits  will  enable  a company  to  make  up  an 
existing  present  deficiency  in  the  legal  net  value.  In  one  or 
more  of  the  States  it  is  expressly  provided  that : 


27 

“ When  the  actual  funds  of  any  life  insurance  company  doing 
business  in  this  Commonwealth  are  not  of  a net  value  equal  to 
its  liabilities,  counting  as  such  the  net  value  of  its  policies 
according  to  the  prescribed  table  of  mortality  and  rate  of  in- 
terest, it  shall  be  the  duty  of  the  Insurance  Commissioner  to 
give  notice  to  such  company  and  its  agents  to  discontinue  issu- 
ing new  policies  within  this  Commonwealth  until  such  time  as 
its  funds  have  become  equal  to  its  liabilities,  valuing  its  policies 
as  aforesaid  ” 

The  fund  which  the  law  designates  as  the  “ net  value  ” of 
the  policies  a company  has  in  force  is  usually  called  “ reserve.” 
In  technical  works  on  life  insurance  this  fund  is  represented  by 
the  symbol  H.  The  English  writers  who  first  used  this  symbol 
explain  that  it  comes  from  the  expression  : 

“ How  much  must  be  in  deposit?” 

In  reference  to  this  fund  one  of  the  leading  actuaries  says: 

u It  does  not  belong  to  the  company.  It  has  been  intrusted 
to  them  for  a specific  object,  for  which  it  ought  to  be  sacredly 
reserved. 

“ The  net  valuation  by  which  the  legal  reserve  is  calculated 
is  proper  and  appropriate  for  many  important  purposes. 

“ It  makes  a just  and  proper  report  of  the  condition  of  the 
company,  judged  by  the  premiums,  losses,  and  expenses  already 
incurred. 

“ It  tells  the  public  if  the  company  has  not  the  legal  reserve, 
that  the  managers  have  dissipated  the  whole  of  their  capital, 
and  that  they  ought  not  to  be  permitted  to  continue  the  busi- 
ness of  insurance  by  making  new  contracts  and  issuing  new 
policies.” 

The  principles  upon  which  calculations  of  legal  net  values 
are  made  not  only  enable  us  to  determine  the  amount  that 
should  be  held  by  the  company  to  the  credit  of  the  policies  it 
has  in  force,  but  these  principles  illustrate  the  fact  that  the  net 
premium  is  composed  of  two  parts — one  of  which  pays  each 
year  the  net  cost  of  insuring  the  amount  at  risk  on  the  policy 
during  that  year,  and  the  other  goes  to  form  the  legal  net  value 
that  must  be  held  by  the  company  for  the  policy.  It  has 
already  been  shown  that  these  separate  parts  of  the  net  premium 
are  susceptible  of  definite  and  easy  computation,  so  that  it 
may  be  known  in  advance,  what  portion  of  the  policyholder’s 
payments  will,  on  the  legal  data,  be  needed  to  insure  the 


28 

amount  at  risk  and  what  portion  will  go  to  form  the  “ reserve  ” 
— as  the  companies  style  the  fund  which  the  law  calls  “ net 
value,”  and  which  the  older  English  writers  considered  to  be  a 
“ deposit”  held  to  the  credit  of  the  policy. 

The  “ loading,”  added  to  net  premiums  for  the  purpose  of 
providing  for  expenses,  profits,  contingencies,  and  so-called 
“ dividends  ” to  policyholders,  is  not  directly  taken  into  consid- 
eration in  net  valuations.  The  simple  elementary  principles 
upon  which  calculations  of  life  insurance  net  values  are  based 
stand  at  the  threshold  of  the  business.  Knowledge  of  these 
principles  should  be  as  common  as  that  of  calculating  interest 
on  money.  Without  clear  conceptions  on  this  subject,  life  in- 
surance cannot  be  understood  any  better  than  the  business  of 
banking  can  be  comprehended  by  one  who  has  no  idea  of  the 
principles  upon  which  calculations  ot  interest  on  money  are 
based.  Yet  it  has  been  recently  said  by  the  highest  authority  : 
“ It  is  wonderful  what  profound  ignorance  prevails  in  reference 
to  the  first  elements  of  this  subject.” 

Officers,  trustees,  and  agents  of  companies,  as  a general  rule, 
have  made  but  little  effort  to  have  this  ignorance  abated  ; on 
the  contrary,  they  seem  more  than  willing  to  have  knowledge 
of  these  simple,  elementary  principles  restricted  to  a few 
actuaries  and  consulting  actuaries  in  the  pay  of  the  companies. 
With  due  deference  to  the  opinions  of  those  able  business  men 
who  have  the  control  of  these  corporations,  it  is  believed  that 
they  make  a great  mistake  in  permitting  knowledge  of  this 
subject  to  be  confined  to  the  “ initiated  few.”  A trust  business, 
involving  such  immense  amounts,  cannot  reasonably  be  ex- 
pected to  permanently  thrive  and  flourish  upon  the  ignorance 
of  the  people. 


I'D 


VI. 


HE  method  by  which  we  determine  the  sum  that  will,  at 


l the  beginning  of  any  year,  be  just  sufficient,  on  the  legal 
data,  to  pay  net  cost  of  insuring  the  amount  at  risk  on  the 
policy  during  that  year,  has  been  explained.  It  is  now  pro- 
posed to  illustrate  in  detail  the  principles  used  in  computing  the 
net  present  value  of  this  sum.  To  do  this — and  at  the  same 
time  give  an  example  showing  how  subjects  of  this  character 
were  discussed  one  hundred  years  ago — the  following  extract  is 
made  from  a work  on  Annuities,  by  Masseres,  published  in  Lon- 
don in  1783.  The  first  problem  he  gives  is,  “ To  find  the  pre- 
sent value  of  a future  sum  of  money  which  is  certainly  to  be 
paid  at  the  end  of  one  or  more  years,  according  to  any  given 
rate  of  interest.”  The  rule  by  which  this  problem  is  solved 
has  already  been  explained.  The  second  problem  given  by  Mas- 
seres  is  that  to  which  attention  is  invited.  He  says : 

“ The  doctrine  of  life  annuities  is  by  no  means  of  so  abstruse 
and  difficult  a nature  as  many  people  are  apt  to  imagine.  A 
moderate  share  of  common  sense,  or  capacity  to  reason  justly, 
and  a knowledge  of  common  arithmetic,  are  all  the  qualities 
that  are  necessary  to  a right  understanding  of  the  principles  on 
which  it  is  founded.” 

He  gives  a table  representing  the  probabilities  of  the  duration 
of  human  life  at  the  several  ages  therein  mentioned,  from  the 
age  of  three  years  to  the  age  of  ninety-five,  grounded  on  lists  of 
the  French  Tontines  or  Long  Annuities,  and  verified  by  a com- 
parison thereof  with  the  necrologies,  or  mortuary  registers,  of 
several  religious  houses  of  both  sexes,  by  M.  de  Parcieux  : 


30 


Age. 

Persons 

Living. 

Age. 

Persons 

Living. 

Age. 

Persons 

Living. 

. Persons 

Age-  Living. 

. Persons 

Age.  Living. 

3 

IOOO 

22 

798 

4* 

650  j 

60 

463 

79 

136 

4 

970 

23 

790 

42 

643 

61 

459 

80 

118 

5 

948 

24 

782 

43 

635 

62 

437 

81 

IOI 

6 

930 

25 

774 

44 

629 

63 

423 

82 

85 

7 

915 

26 

766 

45 

622 

64 

409 

83 

7i 

8 

902 

27 

758 

46 

615 

65 

395 

84 

59 

9 

890 

28 

750 

47 

607 

66 

380 

85 

48 

IO 

880 

29 

742 

48 

599  i 

6 7 

364 

86 

38 

ii 

872 

30 

734 

49 

590  ; 

68 

347 

87 

29 

12 

866 

31 

726 

50 

58i 

69 

329 

88 

22 

13 

860 

32 

718 

5i 

57i 

70 

310 

89 

16 

14 

854 

33 

710 

52 

560 

7i 

291 

90 

11 

15 

848 

34 

702 

53 

549 

72 

271 

9i 

7 

16 

842 

35 

694 

54 

538 

73 

251 

92 

4 

1 7 

a35 

36 

686 

55 

526 

74 

231 

93 

2 

18 

828 

37 

678 

56 

5i4  ' 

75 

211 

94 

1 

19 

821 

38 

671 

57 

502 

76 

192 

95 

0 

20 

814 

39 

664 

58 

489 

77 

173 

21 

8l6 

40 

657 

59 

476 

78 

154 

THE  FUNDAMENTAL  MAXIM  OF  THE  DOCTRINE  OF  LIFE 
ANNUITIES. 

“ In  every  bargain  between  two  persons  concerning  a grant 
of  a sum  of  money  to  be  paid  by  the  one  to  the  other  at  a 
given  future  time,  in  case  the  grantee  or  purchaser  shall  be 
then  alive,  the  fair  price  of  such  future  sum  of  money,  a.ccord- 
ing  to  a given  rate  of  the  interest  of  money  and  a given  table  of 
the  probabilities  of  the  duration  of  human  life,  is  to  be  ascer- 
tained in  the  following  manner:  We  must  suppose,  in  the  first 
place,  that  the  grantor  of  the  future  sum  of  money  makes 
several  hundred  grants  of  the  same  kind,  and  upon  exactly  the 
same  conditions,  to  as  many  different  grantees,  or  purchasers, 
all  of  the  same  age  with  the  first  grantee ; and,  in  the 
second  place,  that  these  several  purchasers  die  off  in  the 
interval  between  the  time  of  making  the  grants  and  the  time 
of  payment,  in  the  same  proportion  as  persons  of  the  same 
ages  respectively  are  represented  to  do  in  the  table  of  the  prob- 
abilities of  the  duration  of  human  life  by  which  the  calculation 
is  to  be  governed,  and,  in  the  third  place,  we  must  suppose  that 
the  several  sums  of  money  paid  by  the  several  grantees  of  these 
future  payments  to  the  grantor  of  them  as  the  price  thereof,  are 
improved  by  the  said  grantor,  at  compound  interest,  at  the  rate 
supposed  in  the  question,  during  the  whole  interval  of  time  be- 
tween the  time  of  making  the  grants  and  the  time  at  which  the 
payments  become  due.  And  then  we  roust  inquire  what  sum 


31 

each  of  the  said  grantees  ought  to  pay  to  the  grantor,  to  the  end 
that,  upon  these  three  suppositions,  he  may,  at  the  end  of  the 
said  interval,  or  when  the  said  payments  become  due,  be  neither 
a gainer  nor  a loser  by  the  sum  total  of  all  his  bargains,  but  be 
possessed  of  just  enough  money,  arising  from  the  sums  form- 
erly paid  him  by  the  said  grantees,  to  satisfy  all  the  demands 
which  will  then  be  made  upon  him.  And  the  sum  which  ought 
thus  to  be  paid  him  by  each  of  the  said  grantees,  when  he 
makes  a great  number  of  said  grants  to  different  persons,  is  the 
fair  price  which  a single  grantee  ought  to  pay  him  for  a grant 
for  the  said  future  sum  of  money,  subject  to  the  same  conditions 
and  contingencies  when  he  makes  only  one  such  grant. 

This  is  a maxim  which,  I presume,  will  be  admitted  as  self- 
evident,  it  being  hardly  possible  to  doubt  of  its  truth.  But  if 
the  reader  should  not  admit  it  upon  its  own  evidence,  I confess 
I am  unable  to  demonstrate  it  by  means  of  any  other  proposi- 
tion more  evident  than  itself.  And,  therefore,  in  this  case,  I 
must  desire  him  to  consider  it  as  a definition  of  what  is  meant 
in  the  following  pages  by  the  expressions  the  ‘ fair  f rice'  or 
‘ true  value  ’ of  such  a future  contingent  payment,  since  it  is 
in  that  sense  that  the  fair  price  or  true  value  of  such  a future 
contingent  payment  can  be  collected  from  the  table  of  the 
probability  of  the  duration  of  human  life  above  described.” 

PROBLEM  II. 

“To  find  the  sum  of  money  which  the  purchaser  of  a future 
payment  of  one  pound  sterling,  to  be  received  at  the  end  of  any 
given  number  of  years,  provided  the  said  purchaser  shall  then 
be  living,  ought  to  pay  for  it — the  age  of  the  said  purchaser, 
and  the  rate  of  interest  of  money,  and  the  probabilities  of  the 
duration  of  human  life,  being  all  given.” 

A SOLUTION  OF  THIS  PROBLEM  IN  THE  CASE  OF  A PAR- 
TICULAR EXAMPLE. 

Let  the  rate  of  interest  of  money  be  supposed  to  be  three  per 
cent,  and  the  probabilities  of  the  duration  of  human  life  such 
as  they  are  represented  to  be  in  Monsieur  de  Parcieux’s  table 
above  mentioned ; and  let  the  number  of  years  at  the  end  of 
which  the  said  sum  of  one  pound  is  to  be  paid  to  the  grantee, 
or  purchaser  of  it,  if  he  be  then  alive,  be  thirty,  and  the  age  of 
the  said  grantee,  or  purchaser,  twenty-five  years. 

Then,  in  the  first  place,  we  must  look  into  M.  de  Parcieux’s 
table  to  see  how  many  persons  of  twenty-five  years  of  age  are 
there  supposed  to  be  all  living  at  the  same  time.  This  number 
we  shall  find  to  be  774.  We  must  therefore  suppose  that  the 


32 

grantor  of  the  one  pound  to  the  purchaser,  proposed  in  the 
question,  does  not  confine  himself  to  that  single  grant,  but 
makes  773  more  such  grants,  of  one  pound  each  to  as  many 
different  persons  of  the  same  age  of  twenty-five  years,  to  be 
paid  to  them  at  the  end  of  thirty  years,  or  when  they  shall  be 
fifty-five  years  old,  if  they  shall  then  be  living,  but  not  to  be 
paid  to  their  executors,  or  other  representatives,  if  they  shall 
then  be  dead ; that  is,  we  must  suppose  that  he  makes  774 
such  grants  in  all,  including  that  of  the  purchaser  proposed  in 
the  question.  And  we  must  likewise  suppose  that  all  these  774 
purchasers  have  the  same  chance,  one  with  the  other,  of  living 
any  given  number  of  years,  or  that  there  is  no  apparent  reason 
for  supposing  that  any  one  of  them  is  more  likely  to  live  to  any 
given  future  age  than  any  other.  This  done,  we  must  inquire 
how  many  of  these  774  purchasers  of  one  pound  each  will  be 
alive  at  the  end  of  thirty  years,  supposing  them  to  die  off  in  the 
proportion  mentioned  in  M.  de  Parcieux’s  table.  Now,  it  ap- 
pears by  M.  de  Parcieux’s  table,  that  out  of  774  persons  of  the 
age  of  twenty-five  years,  all  living  at  the  same  time,  52 6 will  be 
alive  at  the  age  of  fiftv-five  years,  or  at  the  distance  of  thirty 
years.  Therefore,  out  of  the  said  774  purchasers  of  these  future 
payments  of  one  pound,  to  be  received  at  the  end  of  thirty 
years,  526  will  live  to  be  entitled  to  them.  Therefore,  at  the 
end  of  the  said  thirty  years,  the  grantor  of  these  future  pay- 
ments will  have  526  sums,  of  one  pound  each,  to  pay  to  the 
said  surviving  purchasers.  And  consequently,  to  the  end  that 
the  said  grantor  may  be  neither  a gainer  nor  a loser  by  the  sum 
total  of  all  his  bargains,  it  is  necessary  that  he  should  receive  at 
the  time  of  making  the  said  grants  526  times  the  present 
value  of  one  pound,  payable  at  the  end  of  thirty  years,  when 
the  interest  of  money  is  three  per  cent,  or  526  times  the  sum 
which,  being  improved  continually  at  compound  interest  dur- 
ing the  said  term  of  thirty  years  of  the  said  rate  of  interest,  will 
at  the  end  of  that  time  amount  to  one  pound  ; because,  in  that 
case,  if  he  improves  the  said  sum  (of  526  times  the  present 
value  of  one  pound)  so  received,  at  compound  interest,  at  the 
said  rate  of  three  per  cent,  during  the  whole  thirty  years,  it  will 
in  that  time  increase  to  just  526  pounds,  which  is  the  sum  he 
will  then  be  obliged  to  pay  to  the  surviving  purchasers.  The 
present  value  of  one  pound,  payable  at  the  end  of  thirty  years, 
without  being  liable  to  any  contingency,  when  the  interest  of 
money  is  three  per  cent,  is  .41198676  of  a pound.  Therefore, 
526  times  .41198676  of  a pound,  or  £216.70503576,  is  the  sum 
which  the  said  grantor  ought  to  receive,  at  the  time  of  making 
the  said  grants,  from  all  the  774  purchasers  of  them.  There- 
fore, the  sum  which  each  of  them  ought  then  to  pay  him  is  the 


33 

774th  part  of  £216  70503576,  or  .27998066  of  a pound,  or 
nearly  .28  of  a pound,  or  55.  7%d.  And,  consequently,  when  he 
makes  only  one  such  grant  to  a purchaser  of  twenty  years  of  age, 
he  ought  to  receive  for  it  the  same  sum  of  .27998066  of  a pound, 
or  .28  of  a pound,  or  5 s.  >]%d. 

I have  solved  the  foregoing  problem,  in  the  case  of  a particu- 
lar example,  for  the  sake  of  making  the  method  of  solution  as 
clear  and  familiar  as  possible.  But  it  is  easy  to  see  that  the 
reasonings  used  in  it  extend  to  all  other  cases  whatsoever,  and, 
consequently,  that  the  solution  is  really  general.” 

The  above  principles  apply  to  any  table  of  mortality,  to  any 
rate  of  interest  upon  money,  and  to  any  unit  of  value — to  one 
dollar  just  as  well  as  to  one  pound  sterling.  Having  found  the 
net  present  value  of  one  dollar  to  be  paid  at  any  designated 
future  time,  in  case  the  insured  person  is  then  alive,  the  net 
present  value  of  a similar  payment  of  any  other  named  sum 
becomes  known. 

A right  understanding  of  the  principles  upon  which  calcula- 
tions of  life  insurance  legal  net  values  are  made — based  upon  a 
table  of  mortality  and  rate  of  interest  designated  by  law — re- 
quires 44  a knowledge  of  common  arithmetic  ” and  44  a moderate 
capacity  to  reason  justly.”  The  preposterous  claim  that  great 
mathematical  acquirement  is  essential  to  a clear  understanding 
of  the  simple  principles  used  in  making  the  net  calculations 
needed  in  ordinary  life  insurance  is  only  equaled  by  the  credu- 
lity of  those  who  credit  this  absurd  assumption 


